In general if we interpolate a function f x at j

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Unformatted text preview: ::: 1 = J ! dd y ( ): J 0 iJ 21 J The notation is slightly confusing since ( ) denotes an x derivative and the remaining derivatives are taken with respect to . We'll use (9.2.3d) to transfer all derivatives to the physical domain, i.e., d =h d : d dx 0 j Then y 0 i 1 i Taking a vector norm, we have 2 ::: iJ = h( J! y jR(x)j C h J j J +1) ( ): J j which proves the result. Having these preliminary results, we establish a basic stability result. Theorem 9.3.1. The J -stage collocation solution (9.2.7) of linear BVPs (9.2.8) exists and is stable. The discretization error e(x) = y(x) ; y(x) (9.3.7a) satis es max ke( )(x)k = O(h ) 1 j xi ;1 x j = 0 1 ::: J j; J ;j i xi (9.3.7b) where i = h: (9.3.7c) h i Proof. Following the steps leading to (9.2.11), we use (9.3.2) to show that y(x ) = ; y(x 1 ) + g + h : i i i; i i i (9.3.8) The term arises from the E and E terms in (9.3.2) hence, using (9.3.6) it is O(h ). If J 1, the one-step scheme (9.2.7) is consistent. A consistent one-step scheme is also stable and convergent (Theorems 3.4.1, 2). Thus, solutions of (9.2.7) exist and, by subtacting (9.2.11a) from (9.3.8), satisfy i J j i max jy(x ) ; Y...
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